48 research outputs found
An Axiomatic Setup for Algorithmic Homological Algebra and an Alternative Approach to Localization
In this paper we develop an axiomatic setup for algorithmic homological
algebra of Abelian categories. This is done by exhibiting all existential
quantifiers entering the definition of an Abelian category, which for the sake
of computability need to be turned into constructive ones. We do this
explicitly for the often-studied example Abelian category of finitely presented
modules over a so-called computable ring , i.e., a ring with an explicit
algorithm to solve one-sided (in)homogeneous linear systems over . For a
finitely generated maximal ideal in a commutative ring we
show how solving (in)homogeneous linear systems over can be
reduced to solving associated systems over . Hence, the computability of
implies that of . As a corollary we obtain the computability
of the category of finitely presented -modules as an Abelian
category, without the need of a Mora-like algorithm. The reduction also yields,
as a by-product, a complexity estimation for the ideal membership problem over
local polynomial rings. Finally, in the case of localized polynomial rings we
demonstrate the computational advantage of our homologically motivated
alternative approach in comparison to an existing implementation of Mora's
algorithm.Comment: Fixed a typo in the proof of Lemma 4.3 spotted by Sebastian Posu
On the computation of -flat outputs for differential-delay systems
We introduce a new definition of -flatness for linear differential delay
systems with time-varying coefficients. We characterize - and -0-flat
outputs and provide an algorithm to efficiently compute such outputs. We
present an academic example of motion planning to discuss the pertinence of the
approach.Comment: Minor corrections to fit with the journal versio
Toric Ideals, Polytopes, and Convex Neural Codes
How does the brain encode the spatial structure of the external world?
A partial answer comes through place cells, hippocampal neurons which
become associated to approximately convex regions of the world known
as their place fields. When an organism is in the place field of some place
cell, that cell will fire at an increased rate. A neural code describes the set
of firing patterns observed in a set of neurons in terms of which subsets
fire together and which do not. If the neurons the code describes are place
cells, then the neural code gives some information about the relationships
between the place fieldsâfor instance, two place fields intersect if and only if
their associated place cells fire together. Since place fields are convex, we are
interested in determining which neural codes can be realized with convex
sets and in finding convex sets which generate a given neural code when
taken as place fields. To this end, we study algebraic invariants associated
to neural codes, such as neural ideals and toric ideals. We work with a
special class of convex codes, known as inductively pierced codes, and seek
to identify these codes through the Gröbner bases of their toric ideals
Une approche par lâanalyse algĂ©brique effectivedes systĂšmes linĂ©aires sur des algĂšbres de Ore
The purpose of this paper is to present a survey on the effective algebraic analysis approach to linear systems theory with applications to control theory and mathematical physics. In particular, we show how the combination of effective methods of computer algebra - based on Gröbner basis techniques over a class of noncommutative polynomial rings of functional operators called Ore algebras - and constructive aspects of module theory and homological algebra enables the characterization of structural properties of linear functional systems. Algorithms are given and a dedicated implementation, called OreAlgebraicAnalysis, based on the Mathematica package HolonomicFunctions, is demonstrated.Le but de ce papier est de prĂ©senter un Ă©tat de lâart dâune approche par lâanalyse algĂ©brique effective de la thĂ©orie des systĂšmes linĂ©aires avec des applications Ă la thĂ©orie du contrĂŽle et Ă la physique mathĂ©matique.En particulier, nous montrons comment la combinaison des mĂ©thodes effectives de calcul formel - basĂ©es sur lestechniques de bases de Gröbner sur une classe dâalgĂšbres polynomiales noncommutatives dâopĂ©rateurs fonctionnels appelĂ©e algĂšbres de Ore - et dâaspects constructifs de thĂ©orie des modules et dâalgĂšbre homologique permet lacaractĂ©risation de propriĂ©tĂ©s structurelles des systĂšmes linĂ©aires fonctionnels. Des algorithmes sont donnĂ©s et uneimplĂ©mentation dĂ©diĂ©e, appelĂ©e OREALGEBRAICANALYSIS, basĂ©e sur le package Mathematica HOLONOMIC-FUNCTIONS, est prĂ©sentĂ©
On the Rapoport-Zink space for over a ramified prime
In this work, we study the supersingular locus of the Shimura variety
associated to the unitary group over a ramified prime. We
show that the associated Rapoport-Zink space is flat, and we give an explicit
description of the irreducible components of the reduction modulo of the
basic locus. In particular, we show that these are universally homeomorphic to
either a generalized Deligne-Lusztig variety for a symplectic group or to the
closure of a vector bundle over a classical Deligne-Lusztig variety for an
orthogonal group. Our results are confirmed in the group-theoretical setting by
the reduction method \`a la Deligne and Lusztig and the study of the admissible
set
Key problems in the extension of module-behaviour duality
AbstractThe duality for linear constant coefficient partial differential equations between behaviours and finitely generated modules over the operator ring is a very powerful tool linking equation structure to dynamic behaviour. This duality is critically dependent on the choice of signal space. In this paper we discuss two key algebraic problems which form an obstacle to the extension of this theory to general signal spaces. The first of these is the so-called Willems closure problem, which limits the ability of system equations to directly describe the system. The second is the elimination problem, the general solution of which depends upon an algebraic property (injectivity) of the signal space. We demonstrate the importance of these problems in the module-behaviour framework, and some of the useful consequences of a full or partial solution. The issues here are of particular relevance to the extension of the current duality theory for behaviours defined by linear partial differential equations from the case of constant to non-constant coefficients